24 | January | 2008 | A kind of library

116b- Lecture 6

January 24, 2008

We proved that a set is recursive iff it is $\Delta_1$ -definable.

We showed some elementary properties of the theory $\mathsf{Q}$ , defined end-extensions, and verified that $\mathsf{Q}$ is $\Sigma_1$ -complete.

We also defined what it means for a function, or a set, to be represented in a theory.

Leave a Comment » | 116b: Computability and incompleteness | Permalink
Posted by andrescaicedo

You are currently browsing the A kind of library blog archives for the day Thursday, January 24th, 2008.
Categories

Categories
Search

Mathematical links
My pages
Others
Mathoverflow
- Answer by Andrés E. Caicedo for Continuum Hypothesis December 30, 2012
  A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematics of the Polish Academy of Sciences. Sierpiński discusses equivalences and consequences. The statements covered include examples from set theory, combinatorics, […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Which journals publish expository work? October 25, 2013
  There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads: The EMS Surveys in Mathematical Sciences is dedicated to publishing authoritative surveys and high-level expositions in all areas of mathematical sciences. It is a peer-reviewed periodical […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Does "$X \not\to (\omega)^\omega_2$ for every infinite $X$" imply ${\sf AC}$? March 29, 2022
  The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega_2$$ does not imply the axiom of choice. This was shown by Kleinberg and Seiferas in 1973, see MR0340025 (49 #4782) Kleinberg, E. M.; Seiferas, J. I. Infinite exponent partition relations and well-ordered choice. J. Symbolic Logic 38 (1973), 299–308. https://doi.org/10.23 […]
  Andrés E. Caicedo
- On the structure of maximal Ramsey colorings October 25, 2021
  For positive integers $a_1,\dots,a_n$, recall that the multicolor Ramsey number $R(a_1,\dots,a_n)$ is the smallest integer $N$ such that if the edges of the complete graph $K_N$ are colored with the $n$ colors $1,\dots,n$, then there is some $i\le n$ and a set of $a_i$ vertices, all of whose edges received color $i$. A maximal Ramsey$(a_1,\dots,a_n)$-colorin […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial? November 29, 2010
  Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for Are there examples of theorems proved via proper (i.e. non-conservative) extensions? May 7, 2013
  Yes, this is a nice idea, and the approach is used in practice. I list four examples below, but there are many others. Any arithmetic statement, or any first order statement about $(\mathbb R,\mathbb N,+,\times,
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for If a function can only be defined implicitly does it have to be multivalued? June 4, 2011
  Not necessarily. Consider the graph $G$ in ${\mathbb R}^2$ of the points $(x,y)$ such that $$ y^5+16y-32x^3+32x=0. $$ This example comes from the nice book "The implicit function theorem" by Krantz and Parks. Note that this is the graph of a function: Fix $x$, and let $F(y)=y^5+16y-32x^3+32x$. Then $F'(y)=5y^4+16>0$ so $F$ is strictly incre […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Whether the map $x\mapsto x^3$ in a finite field is bijective August 20, 2013
  Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is it possible to formalize the idea that large cardinal axioms don't limit us, while their negations do? November 25, 2013
  One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is $6.12345678910111213141516171819202122\ldots$ transcendental? March 5, 2011
  This is a transcendental number, in fact one of the best known ones, it is $6+$ Champernowne's number. Kurt Mahler was first to show that the number is transcendental, a proof can be found on his "Lectures on Diophantine approximations", available through Project Euclid. The argument (as typical in this area) consists in analyzing the rate at […]
  Andrés E. Caicedo
January 2008

M T W T F S S

1 2 3 4 5 6

7 8 9 10 11 12 13

14 15 16 17 18 19 20

21 22 23 24 25 26 27

28 29 30 31

« Sep Feb »
Archives
- June 2019 (2)
- February 2019 (7)
- December 2018 (1)
- October 2018 (2)
- August 2018 (1)
- February 2018 (1)
- October 2017 (1)
- July 2017 (1)
- September 2016 (2)
- August 2016 (3)
- July 2016 (2)
- April 2016 (1)
- October 2015 (1)
- June 2015 (1)
- May 2015 (1)
- April 2015 (7)
- March 2015 (5)
- February 2015 (7)
- January 2015 (11)
- December 2014 (2)
- November 2014 (5)
- October 2014 (6)
- September 2014 (3)
- August 2014 (3)
- April 2014 (3)
- March 2014 (5)
- February 2014 (2)
- January 2014 (2)
- December 2013 (2)
- November 2013 (10)
- October 2013 (6)
- September 2013 (7)
- August 2013 (5)
- June 2013 (1)
- May 2013 (8)
- April 2013 (7)
- March 2013 (5)
- February 2013 (9)
- January 2013 (2)
- November 2012 (1)
- September 2012 (1)
- August 2012 (3)
- May 2012 (3)
- April 2012 (10)
- March 2012 (4)
- February 2012 (13)
- January 2012 (15)
- November 2011 (8)
- October 2011 (14)
- September 2011 (7)
- August 2011 (1)
- April 2011 (1)
- March 2011 (3)
- February 2011 (4)
- January 2011 (7)
- December 2010 (1)
- November 2010 (7)
- October 2010 (12)
- September 2010 (13)
- August 2010 (3)
- May 2010 (4)
- April 2010 (11)
- March 2010 (6)
- February 2010 (12)
- December 2009 (9)
- November 2009 (11)
- October 2009 (13)
- September 2009 (19)
- August 2009 (10)
- June 2009 (1)
- May 2009 (2)
- April 2009 (14)
- March 2009 (17)
- February 2009 (19)
- January 2009 (15)
- December 2008 (5)
- November 2008 (12)
- October 2008 (17)
- September 2008 (10)
- August 2008 (3)
- June 2008 (3)
- May 2008 (15)
- April 2008 (21)
- March 2008 (10)
- February 2008 (13)
- January 2008 (11)
- September 2007 (1)
- July 2007 (1)
- March 2007 (4)
- February 2007 (20)
- January 2007 (23)
- December 2006 (4)
Alan Turing Alexander S. Kechris Alfred Tarski algebraic numbers Andras Hajnal Andreas Blass Andrew Michael Bruckner arithmetic mean-geometric mean inequality Arnoud C. M. van Rooij Augustin Cauchy Axel Thue Bernhard Riemann BEST bpfa compactness completeness covering property determinacy Donald A. Martin elementary embedding Emile Borel Euclidean algorithm extension field field field extension Francisco Frank Plumpton Ramsey Fred Galvin Galvin-Hajnal theorem GCH Georg Cantor Godfrey Harold Hardy Grigor Sargsyan Harvey Friedman ideal Isaac Newton James Baumgartner Jerome Keisler Johan Thim John R. Steel Karl Theodor Wilhelm Weierstrass Koenig's lemma Kurt Goedel Lech Maligranda Marion Scheepers measurable cardinal Menachem Magidor midterm mm Nowhere differentiable function Paul Erdos Paul Larson pfa propositional logic quartic equation Ramsey theorem Rene Baire Richard Rado Robert Solovay Saharon Shelah Samuel Coskey sch Silver's theorem Stefan Banach Stevo Todorcevic supercompact cardinals Tarski's conjecture Turing degrees ultrafilter ultrapower Undecidability and incompleteness Unsolvable problems W. Hugh Woodin Wacław Franciszek Sierpiński Wilhelmus Hendricus Schikhof

	Is there a Borel-mea… on 414/514 Examples of Baire clas…
	Simple bijectiveness… on 580 -Some choiceless results…
	Schur’s theore… on 580 -III. Partition calcu…
	Uncountable linearly… on 403/503 – Homework …
	What is a non-constr… on On strong measure zero se…
	function from Nmathb… on Hindsight
	Cauchy Schwarz Inequ… on 275 -Positive polynomials
	What is a Cantor-sty… on 580 -Some choiceless results…
	Question about Gener… on 580 -Some choiceless results…
	Set has Measure Zero… on On strong measure zero se…
	Non-measurable subse… on 580 -Cardinal arithmetic …
	hmeng1 on 414/514 The theorems of Rieman…
	580 -Cardinal arithm… on 580 -Cardinal arithmetic …
	580 -Cardinal arithm… on 580 -Cardinal arithmetic …
	580 -Cardinal arithm… on 580 -Cardinal arithmetic …

A kind of library

116b- Lecture 6

Categories

Recent Comments

Mathematical links

My pages

Others

Mathoverflow

Math.StackExchange

Archives